Abstract
In recent years, various operators of fractional calculus (that is, calculus of integrals and derivatives of arbitrary real or complex orders) have been investigated and applied in many remarkably diverse fields of science and engineering. Many authors have demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. The purpose of this paper is to present a certain class of the explicit particular solutions of the associated CauchyEuler fractional partial differential equation of arbitrary real or complex orders and their applications as follows:
where ; A, B, C, M, N, α and β are arbitrary constants.
MSC: 26A33, 33C10, 34A05.
Keywords:
fractional calculus; differential equation; partial differential equation; generalized Leibniz rule; analytic function; index law; linearity property; principle value; initial and boundary valueDedication
Dedicated to Professor Hari M Srivastava.
1 Introduction, definitions and preliminaries
The subject of fractional calculus (that is, derivatives and integrals of any real or complex order) has gained importance and popularity during the past two decades or so, due mainly to its demonstrated applications in numerous seemingly diverse fields of science and engineering (cf.[115]). By applying the following definition of a fractional differential (that is, fractional derivative and fractional integral) of order , many authors have obtained particular solutions of a number of families of homogeneous (as well as nonhomogeneous) linear fractional differintegral equations.
In this paper, we present a direct way to obtain explicit solutions of such types of the associated CauchyEuler fractional partial differential equation with initial and boundary values. The results are a coincidence that the solutions are obtained by the methods applying the Laplace transform with the residue theorem. In this paper, we present some useful definitions and preliminaries for the paper as follows.
If the function is analytic and has no branch point inside and on , where
is an integral curve along the cut joining the points z and , is an integral curve along the cut joining the points z and ,
and
and
then () is said to be the fractional derivative of of order ν and () is said to be the fractional integral of of order −ν, provided that
First of all, we find it is worthwhile to recall here the following useful lemmas and properties associated with the fractional differintegration which is defined above.
Lemma 1.1 (Linearity property)
If the functionsandare singlevalued and analytic in some domain, then
Lemma 1.2 (Index law)
If the functionis singlevalued and analytic in some domain, then
Lemma 1.3 (Generalized Leibniz rule)
If the functionsandare singlevalued and analytic in some domain, then
whereis the ordinary derivative ofof ordern (), it being tacitly assumed (for simplicity) thatis the polynomial part (if any) of the product.
Lemma 1.4 (Cauchy’s residue theorem)
Let Ω be a simple connected domain, and letCbe a simple closed positively oriented contour that lies in Ω. Iffis analytic insideCand on C, expect at the pointthat lie insideC, then
(I) Iffhas a simple pole at, then
(II) Iffhas a pole of orderkat, then
For a constanta,
Proof The proofs between ‘ν is not an integer’ and ‘ν is an integer’ are not coincident, so we mention the proof as follows.
Therefore we have Property 1.1 for arbitrary . □
Property 1.2For a constanta,
Property 1.3For a constanta,
The fractional derivative of a causal functionis defined by
wheredenotes the ordinary derivative of ordernand Γ is the gamma function.
The Laplace transform of a functionis denoted as
wheresis the Laplace complex parameter. We recall from the fundamental formula (cf. [16])
2 Main results
Theorem 2.1The fractional partial differential equation
with, , andA (≠0), B, C, M, Nare constants, has its solutions of the form given by
(a)
(b)
when the discriminant, and the roots, of Equation (2.5) are repeated; that is, ;
(c)
when the discriminant, and, are the conjugate pair roots of Equation (2.5).
So that the given equation (2.1) becomes
Equation (2.4) leads to the auxiliary equation
That is,
and
are the two roots of the auxiliary equation (2.5). Thus, is a solution of the fractional partial differential equation (2.1) whenever () is a solution of the auxiliary equation (2.5).
There are three different cases to be considered, depending on whether the roots of this quadratic equation (2.5) are distinct real roots, equal real roots (repeated real roots), or complex roots (roots appear as a conjugate pair). The three cases are due to the discriminant of the coefficients .
• Case I: Distinct real roots (when ).
Let and denote the real roots of Equation (2.5) such that . Then the general solution of Equation (2.1) is
• Case II: Repeated real roots (when ).
If the roots of Equation (2.5) are repeated, that is, , then the general solution of Equation (2.1) is
• Case III: Conjugate complex roots (when ).
If the roots of Equation (2.5) are the conjugate pair and , then a solution of Equation (2.1) is
In general,
forms a fundamental solution, where , and λ are constants. □
Theorem 2.2The fractional partial differential equation
with, , andA (≠0), B, C, M, Nare constants, has its solutions of the form given by
(a′)
(b′)
when the discriminant, and the roots, of Equation (2.10) are repeated; that is, ;
(c′)
when the discriminant, and, are the conjugate pair roots of Equation (2.10).
Proof The similarity between the forms of solutions of Equation (2.1) and solutions of a linear equation with constant coefficients of Equation (2.6) is not just a coincidence.
So that the given equation (2.6) becomes
If and are the two roots of the auxiliary equation
then
and
The analysis of three cases is similar to Theorem 2.1, we can obtain each solution of the forms as follows:
with , two distinct real roots,
with , repeated real roots, and
with the conjugate complex roots.
□
Remark The constant λ in Equations (2.2) and (2.7) can be solved directly by constant initial value and constant boundary values (or by the numerical methods).
Corollary 2.1The fractional partial differential equation
with, , andA (≠0), B, C, Mare constants, has its solutions of the form given by
(a″)
(b″)
when the discriminant, and the roots, of Equation (2.5) withare repeated; that is, ;
(c″)
when the discriminant, and, are the conjugate pair roots of Equation (2.5) with.
Corollary 2.2The fractional partial differential equation
with, , andA (≠0), B, C, Mare constants, has its solutions of the form given by
when the discriminant, and the roots, of Equation (2.10) withare repeated; that is, ;
when the discriminant, and, are the conjugate pair roots of Equation (2.10) with.
3 Examples
Example 3.1 If the twodimensional harmonic equation is transformed to plane polar coordinates r and θ, defined by , , it takes the form
then it has solutions of the form
Solution Equation (3.1) is coincident to
We have the solution
by taking , , , , and in Theorem 2.1. □
Example 3.2 The fractional partial differential equation
Solution Putting , , and in Corollary 2.2, we obtain the solution
The analysis of the case is similar to Theorem 2.2. □
Example 3.3 The fractional partial differential equation
Solution Putting , , and and in Corollary 2.1, we obtain the solution
Then
If the discriminant , the solution is trivial. If the discriminant , then the solution is
□
Example 3.4 The fractional partial differential equation
Solution Putting , , , and in Corollary 2.2, the discriminant is , but leads to a contradiction, hence there are different real roots and , so that we have
By the boundary condition , we obtain and . So,
and the particular solution is
If we apply the Laplace transform to , then and . Using the residue theorem,
The solution obtained by the method of Laplace transform and the residue theorem is a coincidence, which is our result above. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
SDL carried out the molecular genetic studies, participated in the sequence alignment and drafted the manuscript. CHL carried out the immunoassays. SMS participated in the sequence alignment.
Acknowledgements
The authors are deeply appreciative of the comments and suggestions offered by the referees for improving the quality and rigor of this paper. The present investigation was supported, in part, by the National Science Council of the Republic of China under Grant NSC1012115M033002.
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